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Ask HN: How to self-study mathematics from the undergrad through graduate level?

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Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#191

Post by School-Cotton »

momofarm wrote: post #189
School-Cotton wrote: post #163

Earlier quoted context omitted.

Math is not just a tool; it is an area of intellectual exploration. By the same logic studying history or philosophy is a colossal waste of time, too.

so, in real world application, what's the most useful/valuable topic in mathematics? Although I wonder for 90% people of this world that math is just a tool to pass the exam at school, not any real application (or they just can't sense it).

> in real world application, what's the most useful/valuable topic in mathematics?

Basic arithmetic: addition, subtraction, multiplication, division.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#192

Post by aoki »

oefrha wrote: post #169
alnar wrote: post #105

SOME ADVICE BEFORE YOU DO A DEEP DIVE INTO WHATEVER YOU END UP STUDYING MATH-WISE: My Background: Current Undergraduate in CS and I recently added Mathematics The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on…

> The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. I’m afraid you haven’t delved into any advanced topics. That’s actually about the easiest part, and could be mastered by ten year olds (certainly myself when I was ten).

alnar is likely referring to the US system, or something like it: unless you are tutored externally (rich), an autodidact outlier (gifted), or selected for honors courses, you basically take computational courses for 14 years (with one cursory stop for euclidean geometry) and are then thrown into proofs at the age of 19-20, if at all. it's widely recognized as a problem in the math pipeline, which is why many US universities have "transition" courses for non-honors students.

so it's not that the basics are intellectually difficult as much as practically difficult (unfamiliar, disorienting) for many students. many "transition" books talk about the difficulty in adjusting from talent being redefined from perfectionist "plug and chug" (APs, SATs) to reasoning and creativity.

btw, i'm impressed that you could master college-level proofs at 10. i have a kid about that age who is pretty good at logical reasoning, but i'm not sure what topic (at that level) he could do a rigorous proof about; maybe numbers, as in landau? can you say more about the materials you used?

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#193

Post by oefrha »

aoki wrote: post #186
oefrha wrote: post #166

Earlier quoted context omitted.

I have to disagree, especially about abstract algebra. Most concepts and theorems feel like abstract nonsense (not specifically talking about category theory here) when you don’t understand them, but should become pretty natural once you do. For true mastery you need to work with the concepts and results on a day to day basis for a while, by applying them; continually reading the text of definitions and theorems hard…

i agree that memorization by internalization (concepts) is different from (and superior to) memorization by rote (text); i was agreeing with the other fellow that rote memorization is not the end goal. however, to me they're both "memorization" because they both represent work to achieve fluency in application. however, i made my comment because i think i disagree that spaced repetition has no place in learning. i th…

When your definition of memorization encompasses all forms of learning, saying memorization is crucial to learning is pretty much a tautology. No one claimed that amnesia sufferers are perfectly good for mathematics. Same goes for repetition. No one expects you to understand and never forget the theory of cohomology by writing down a long exact sequence once.

The thing is, the root comment of this thread specifically talks about "continually review what you've learned and NEVER forget anything" through looking at highlighted notes in the software mentioned (a somewhat out-of-place plug, I'd say). That's not how math works. You refresh your memory by tackling preferably new problems. Reciting proofs is largely pointless (except for certain very elegant proofs, in which case you probably won't need to recite them anyway); reciting definitions and theorems is even less useful.

> it's meant to 'freeze' all of your knowledge so that you never forget it - ever.

Yeah, no, you don't "freeze" your mathematical knowledge.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#194

Post by thisiszilff »

You want to learn more about both pure and applied mathematics. Pure math is very proof heavy which is excellent for self learning because you never have to take anything on faith. Proofs are meant to be very clear explanations.

One of the best resources for learning in this way is to find books that are structured to have you develop the theory via the problem sets. Of the top of my head, Tao's Analysis books and Atiyah & MacDonald's Commutative Algebra book are exemplars of this style. These books often show the basic definitions, some results, and then break up remaining results into step by step problems.

Proofs will be the hardest part to wrap your head around. Once you understand proofs, can read them, the remaining is going to be a question of time. Think of it like programming: you can't learn by reading, lectures, etc. You need to learn by doing. In this case, the doing is doing proofs.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#195

Post by DoreenMichele »

I was a homeschooling parent. I also have a stronger math background than average, though it isn't anything impressive for this crowd.

I will suggest you try to find a written curriculum from a respectable source as your first step so you have some idea what you need to cover to meet the stated goal.

Then you will want a variety of study materials for the subjects in question. You will want to verify that these are respected materials that are not full of errors.

When I brushed up on math to CLEP Alegebra so I could take college statistics, the computer program I used for self study had errors in it. I knew enough math that I knew what was wrong. I was just rusty and in need of a bit of practice. But it was a program I did not use with my sons to homeschool them because of the errors.

You want to know which materials math geeks like. You want them vetted, basically.

Then use whatever quality materials most appeal to you. Different strokes for different folks and self study let's you pursue whichever materials you like the best.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#196

Post by analog31 »

joe_the_user wrote: post #118
abnry wrote: post #38

Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don…

The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in. I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and ga…

>>> I would guess that each person has a somewhat unique motivation strategy.

Indeed, speaking only for myself, I don't think that I had the self discipline to carry through with such a plan as a teenager. I needed the classroom environment to learn math & physics, which became my college majors. At the same time, I was able to teach myself programming and electronics quite easily for some mysterious reason. And I ended up combining all of those things in grad school.

Also, I only had access to the books and resources of a typical suburb with a decidedly anti-intellectual culture. The officials at my high school refused to offer a calculus course because they said it would be "elitist."

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#197

Post by susam »

Karrot_Kream wrote: post #139
susam wrote: post #117

Earlier quoted context omitted.

Can we start something like this online for Hacker News community who are interested in Mathematics? I have created a group here if someone is interested to join: https://groups.google.com/d/forum/projectfermat (If you think that there is a better place to have a forum like this that anyone can easily view or participate in, please let us know. I mean we could also create an IRC channel, Slack workspace, etc. but the…

I'd be very interested in something like this.

Glad to know you would be interested in this. Please feel free to click on group link I have shared in my previous comment and hit "Apply to join group".

There are 12 members in the group so far which I believe is a good size to kick-start the discussions and other activities. More members are welcome!

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#198

Post by oskkejdjdkjd »

Go to the library and read their math books. I don’t know why the internet is filled with these questions that are always “how do I start doing x?” Just f* do it dude. Nobody is going to be able to or bother to answer anyway. The answers wouldn’t make deep sense to you because they rely on having done it before. Once you’ve started and you’re really stuck and everything you read doesn’t seem to illuminate it, post a question. People will notice you’ve put some work and time in and will be eager to help you.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#199

Post by fromMars »

forkandwait wrote: post #57
abnry wrote: post #38

Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don…

Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.

I'm with you but I think it depends on the person.

I have actually found people to be very different in this regard.

For instance, I work in data science and a lot of my peers like to learn about new techniques by applying them to real problems or working with datasets and exploring.

I don't like that approach. I always like to learn the theory of something before using it.

Similarly, I had the goal of learning math for statistics and general relativity among other subjects.

But my desire to have a deep understanding, ended up with me essentially learning the equivalent an undergraduate math degree and selected graduate level topics.

In my case, just learning in a bottom up way with maybe a slight direction would have been sufficient.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#200

Post by fromMars »

tprice7 wrote: post #146
forkandwait wrote: post #57

Earlier quoted context omitted.

Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.

Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid. Links: http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508... Look for my name (Thomas / Tom Price) on these pages: https://web.archive.org/web…

Just curious, were you working while you studied? How did you manage this?
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